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On the representability of actions of Leibniz algebras and Poisson algebras
Part of
Alan S. Cigoli, Manuel Mancini, Giuseppe Metere
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023

Published online by Cambridge University Press:

22 November 2023, pp. 998-1021
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In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

1 - An Introduction to Affine Lie Algebras and the Associated Groups
Shrawan Kumar, University of North Carolina, Chapel Hill
Book:

Conformal Blocks, Generalized Theta Functions and the Verlinde Formula

Published online:

19 November 2021

Print publication:

25 November 2021, pp 1-67
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Summary

The aim of this chapter is first to set some basic notation and preliminaries (to be used through the book) and then recall the definition of affine Kac--Moody Lie algebras and their basic representation theory and to study the associated groups and their flag varieties. We define the loop group G((t)) (without the central extension) and its various subgroups. Then, we study the associated infinite Grassmannian and prove that it is a reduced ind-variety when G is a semisimple group. We show that G((t)) is a reduced affine ind-scheme. We further study the central extensions of G((t)).

On Mennicke Groups of Deficiency Zero II
Muhammad A. Albar, Abdul-Aziz A. Al-Shuaibi
Journal:

Canadian Mathematical Bulletin / Volume 34 / Issue 3 / 01 September 1991

Published online by Cambridge University Press:

20 November 2018, pp. 289-293

Print publication:

01 September 1991
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Let M be the group defined by the presentation 〈 x, y, z | xy = xm, yz = yn, zx = zr〉,m,n,r ∊ Z. M is one of the few 3-generator finite groups of deficiency zero. These groups have been considered by Mennicke [3], Macdonald, Wamsley [10], Johnson and Robertson [7], and Albar. Properties like the order of M, the nilpotency and solvability were studied. In this paper we give a better upper bound for M than the one given by Johnson and Robertson [7]. We also describe the structure of some cases of M.